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Properties

Label 4-2106e2-1.1-c1e2-0-21
Degree $4$
Conductor $4435236$
Sign $1$
Analytic cond. $282.794$
Root an. cond. $4.10079$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·5-s − 4·7-s − 8-s − 2·10-s − 2·11-s − 13-s − 4·14-s − 16-s − 8·17-s + 14·19-s − 2·22-s + 5·25-s − 26-s + 9·29-s + 10·31-s − 8·34-s + 8·35-s + 14·37-s + 14·38-s + 2·40-s − 5·41-s + 2·43-s + 47-s + 7·49-s + 5·50-s + 28·53-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.894·5-s − 1.51·7-s − 0.353·8-s − 0.632·10-s − 0.603·11-s − 0.277·13-s − 1.06·14-s − 1/4·16-s − 1.94·17-s + 3.21·19-s − 0.426·22-s + 25-s − 0.196·26-s + 1.67·29-s + 1.79·31-s − 1.37·34-s + 1.35·35-s + 2.30·37-s + 2.27·38-s + 0.316·40-s − 0.780·41-s + 0.304·43-s + 0.145·47-s + 49-s + 0.707·50-s + 3.84·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4435236 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4435236 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4435236\)    =    \(2^{2} \cdot 3^{8} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(282.794\)
Root analytic conductor: \(4.10079\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4435236,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.407493605\)
\(L(\frac12)\) \(\approx\) \(2.407493605\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
13$C_2$ \( 1 + T + T^{2} \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_ab
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.e_j
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_ah
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.17.i_by
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.19.ao_dj
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.23.a_ax
29$C_2^2$ \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.29.aj_ca
31$C_2^2$ \( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.31.ak_cr
37$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.37.ao_et
41$C_2^2$ \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.41.f_aq
43$C_2^2$ \( 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.43.ac_abn
47$C_2^2$ \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \) 2.47.ab_abu
53$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.53.abc_lq
59$C_2^2$ \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.59.ai_f
61$C_2^2$ \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.61.k_bn
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_abz
71$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.71.ac_fn
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.73.i_gg
79$C_2^2$ \( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.79.ah_abe
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.83.a_adf
89$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \) 2.89.abi_rz
97$C_2^2$ \( 1 - 8 T - 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.97.ai_abh
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.217363009221607622289043225361, −8.956237102199977924745193846793, −8.582304271517619765486397736477, −8.079134573172441476662088415698, −7.52022101883778749328398566583, −7.48524587474916363408152770202, −6.81380087535178246794329338655, −6.49348836425518200646706856707, −6.33979722647116035954152713495, −5.62890752207643312051478471711, −5.18073156231626039577222492067, −4.94291402519991589898566216370, −4.24726580334109695552477378505, −4.16067747267020068702742678731, −3.46659938431760266300787535523, −2.95794797490787274068886471879, −2.78172562610511434794653792167, −2.35177840087060277394174382227, −0.809488931553020002777156286872, −0.73185929127315667260526539446, 0.73185929127315667260526539446, 0.809488931553020002777156286872, 2.35177840087060277394174382227, 2.78172562610511434794653792167, 2.95794797490787274068886471879, 3.46659938431760266300787535523, 4.16067747267020068702742678731, 4.24726580334109695552477378505, 4.94291402519991589898566216370, 5.18073156231626039577222492067, 5.62890752207643312051478471711, 6.33979722647116035954152713495, 6.49348836425518200646706856707, 6.81380087535178246794329338655, 7.48524587474916363408152770202, 7.52022101883778749328398566583, 8.079134573172441476662088415698, 8.582304271517619765486397736477, 8.956237102199977924745193846793, 9.217363009221607622289043225361

Graph of the $Z$-function along the critical line